In this paper,we study local rings from the perspective of reverse mathematics.We define local rings in a first-order way by usingΠ_2~0 properties of invertible elements,where for a ring R possibly not commutative,R ...In this paper,we study local rings from the perspective of reverse mathematics.We define local rings in a first-order way by usingΠ_2~0 properties of invertible elements,where for a ring R possibly not commutative,R is left(resp.right)local if for any non-left(resp.non-right)invertible elements x,y∈R,x+y is not left(resp.right)invertible;R is local if for any non-invertible elements x,y∈R,x+y is not invertible.Firstly,we solve a question of Sato on characterizations of commutative local rings in his Ph D thesis(Question 6.22 in Sato(2016))and prove that the statement“a commutative ring is local if and only if it has at most one maximal ideal”is equivalent to ACA_0 over RCA_0.We also obtain a nice corollary in computable mathematics,i.e.,there is a computable non-local ring with exactly two maximal ideals such that each of them Turing computes the Halting set K.Secondly,we study the equivalence among left local rings,right local rings,and local rings,showing that these three kinds of first-order local rings are equivalent over the weak basis theory RCA_0.Finally,we extend the results of reverse mathematics on commutative local rings to noncommutative rings.展开更多
In this paper, a necessary and sufficient condition is given for a commutative Artinian local ring whose annihilating-ideal graph is a star graph. Also, a complete char- acterization is established for a finite local ...In this paper, a necessary and sufficient condition is given for a commutative Artinian local ring whose annihilating-ideal graph is a star graph. Also, a complete char- acterization is established for a finite local ring whose annihilating-ideal graph is a star graph.展开更多
We are interested in studying when the class of local modules is Baer- Kaplansky. We provide an example showing that even over a commutative semisimple ring R, we can find two non-isomorphic simple R-modules S1 and S2...We are interested in studying when the class of local modules is Baer- Kaplansky. We provide an example showing that even over a commutative semisimple ring R, we can find two non-isomorphic simple R-modules S1 and S2 such that the rings EndR（S1） and EndR（S2） are isomorphic. We show that over any ring R, the class of semisimple R-modules is Baer Kaplansky if and only if so is the class of simple R-modules.展开更多
Let R-Fpm+uFpm+vFpm+uvFpm,where u2=v2=0,uv=vu.Then R is a local ring,but it is not a chain ring.R contains precisely(pm-1)p3 m units,namely,α+uβ+vγ+uvδ,where α,β,γ,δ∈Fpm,α≠0.In this paper,we investigate all...Let R-Fpm+uFpm+vFpm+uvFpm,where u2=v2=0,uv=vu.Then R is a local ring,but it is not a chain ring.R contains precisely(pm-1)p3 m units,namely,α+uβ+vγ+uvδ,where α,β,γ,δ∈Fpm,α≠0.In this paper,we investigate all constacyclic codes of length ps over R.Firstly,we classify allα-constacyclic and(α+uvβ)-constacyclic codes of length ps over R,respectively,and obtain their structure in each of thoseα-constacyclic and(α+uvβ)-constacyclic codes.Secondly,we address the(α+uβ)-constacyclic codes of length ps over R,and get their classification and structure.Finally,using similar discussion of(α+uβ)-constacyclic codes,we obtain the classification and the structure of α+vβ,α+uβ+uvγ,α+vβ+uvγ,α+uβ+vγ,α+uβ+vβ+uvδ-constacyclic codes of length ps over R.展开更多
We study the algebraic structure of rings R whose zero-divisor graph T(R)has clique number four.Furthermore,we give complete characterizations of all the finite commutative local rings with clique number 4.
We study skew cyclic codes over a class of rings R=F0■F1■⋯■Ft−1,where each Fi(i=0,…,t−1)is a finite field.We prove that a skew cyclic code of arbitrary length over R is equivalent to either a usual cyclic code or ...We study skew cyclic codes over a class of rings R=F0■F1■⋯■Ft−1,where each Fi(i=0,…,t−1)is a finite field.We prove that a skew cyclic code of arbitrary length over R is equivalent to either a usual cyclic code or a quasi-cyclic code over R.Moreover,we discuss possible extension of our results in the more general setting ofδR-dual skew constacyclic codes over R,whereδR is an automorphism of R.展开更多
1. Introduction Since O’Meara worked out the automorphisms of orthogonal groups Ωn(V) over fields in [4] by using well-known residual space method, many results about the isomorphisms and automorphisms of orthogonal...1. Introduction Since O’Meara worked out the automorphisms of orthogonal groups Ωn(V) over fields in [4] by using well-known residual space method, many results about the isomorphisms and automorphisms of orthogonal groups over integral domains have been achieved. Refer to O’Meara, Hahn for example. B. R. McDonald, in [12], determined the automorphisms of O(V) over local rings with 2 a unit by using involutions.展开更多
基金supported by National Natural Science Foundation of China(Grant No.12301001)。
文摘In this paper,we study local rings from the perspective of reverse mathematics.We define local rings in a first-order way by usingΠ_2~0 properties of invertible elements,where for a ring R possibly not commutative,R is left(resp.right)local if for any non-left(resp.non-right)invertible elements x,y∈R,x+y is not left(resp.right)invertible;R is local if for any non-invertible elements x,y∈R,x+y is not invertible.Firstly,we solve a question of Sato on characterizations of commutative local rings in his Ph D thesis(Question 6.22 in Sato(2016))and prove that the statement“a commutative ring is local if and only if it has at most one maximal ideal”is equivalent to ACA_0 over RCA_0.We also obtain a nice corollary in computable mathematics,i.e.,there is a computable non-local ring with exactly two maximal ideals such that each of them Turing computes the Halting set K.Secondly,we study the equivalence among left local rings,right local rings,and local rings,showing that these three kinds of first-order local rings are equivalent over the weak basis theory RCA_0.Finally,we extend the results of reverse mathematics on commutative local rings to noncommutative rings.
基金The first author is supported by Fundamental Research Funds for the Central Universi- ties （No. XDJK2013C060）, Chongqing Research Program of Application Foundation and Advanced Technology （No. cstc2014jcyjA00028） and Scientific Research Foundation for Doctors of Southwest University （No. SWUl12054）. The second author is supported by National Natural Science Foundation of China （No. 11271250）.
文摘In this paper, a necessary and sufficient condition is given for a commutative Artinian local ring whose annihilating-ideal graph is a star graph. Also, a complete char- acterization is established for a finite local ring whose annihilating-ideal graph is a star graph.
文摘We are interested in studying when the class of local modules is Baer- Kaplansky. We provide an example showing that even over a commutative semisimple ring R, we can find two non-isomorphic simple R-modules S1 and S2 such that the rings EndR（S1） and EndR（S2） are isomorphic. We show that over any ring R, the class of semisimple R-modules is Baer Kaplansky if and only if so is the class of simple R-modules.
基金Supported by Research Funds of Hubei Province(D20144401,Q20174503)。
文摘Let R-Fpm+uFpm+vFpm+uvFpm,where u2=v2=0,uv=vu.Then R is a local ring,but it is not a chain ring.R contains precisely(pm-1)p3 m units,namely,α+uβ+vγ+uvδ,where α,β,γ,δ∈Fpm,α≠0.In this paper,we investigate all constacyclic codes of length ps over R.Firstly,we classify allα-constacyclic and(α+uvβ)-constacyclic codes of length ps over R,respectively,and obtain their structure in each of thoseα-constacyclic and(α+uvβ)-constacyclic codes.Secondly,we address the(α+uβ)-constacyclic codes of length ps over R,and get their classification and structure.Finally,using similar discussion of(α+uβ)-constacyclic codes,we obtain the classification and the structure of α+vβ,α+uβ+uvγ,α+vβ+uvγ,α+uβ+vγ,α+uβ+vβ+uvδ-constacyclic codes of length ps over R.
基金This research was supported by the National Natural Science Foundation of China(No.11801356,No.11401368,No.11971338)by the Natural Science Foundation of Shanghai(No.19ZR1424100).
文摘We study the algebraic structure of rings R whose zero-divisor graph T(R)has clique number four.Furthermore,we give complete characterizations of all the finite commutative local rings with clique number 4.
基金This work was supported by the Ministry of Education and Training of Vietnam(Thai Nguyen University)under Grant No.B2019-TNA-02.
文摘We study skew cyclic codes over a class of rings R=F0■F1■⋯■Ft−1,where each Fi(i=0,…,t−1)is a finite field.We prove that a skew cyclic code of arbitrary length over R is equivalent to either a usual cyclic code or a quasi-cyclic code over R.Moreover,we discuss possible extension of our results in the more general setting ofδR-dual skew constacyclic codes over R,whereδR is an automorphism of R.
文摘1. Introduction Since O’Meara worked out the automorphisms of orthogonal groups Ωn(V) over fields in [4] by using well-known residual space method, many results about the isomorphisms and automorphisms of orthogonal groups over integral domains have been achieved. Refer to O’Meara, Hahn for example. B. R. McDonald, in [12], determined the automorphisms of O(V) over local rings with 2 a unit by using involutions.
